Church and Curry: Combining Intrinsic

and Extrinsic Typing

Frank Pfenning

Dedicated to Peter Andrewson the occasion of his retirement

Department of Computer ScienceCarnegie Mellon University

April 5, 2012

1 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classicalAs a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classicalAs a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classicalAs a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classical

As a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classicalAs a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classicalAs a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Church’s Simple Theory of Types

Language and logic for the formalization of mathematics

(Church 1940)Stood the test of time (72 years!)HOL, Isabelle/HOL, TPS, LEO, Satallax, . . .

Synthesis, simplification, and generalization of

Russell and Whitehead’s ramified theory of typesChurch and Rosser’s (untyped) λ-calculus

Some objections

As a computer scientist: classicalAs a philosopher: impredicative

Components

Simply typed λ-calculus (this talk)Logical axioms and inference rules

2 / 24

Simply Typed λ-Calculus

Church’s definitions

Types

1 ι and o are types.2 If α and β are types, then α→ β is a type.

(Church wrote βα)

Well-formed terms Mα of type α

1 Any variable xα or constant cα is a term.2 If xα is a variable and Mβ a term then (λx .M)α→β is a

term.3 If Mα→β

1 and Mα2 are terms, then (M1M2)β is a term.

3 / 24

Simply Typed λ-Calculus

Church’s definitions

Types

1 ι and o are types.2 If α and β are types, then α→ β is a type.

(Church wrote βα)

Well-formed terms Mα of type α

1 Any variable xα or constant cα is a term.2 If xα is a variable and Mβ a term then (λx .M)α→β is a

term.3 If Mα→β

1 and Mα2 are terms, then (M1M2)β is a term.

3 / 24

Simply Typed λ-Calculus

Church’s definitions

Types

1 ι and o are types.2 If α and β are types, then α→ β is a type.

(Church wrote βα)

Well-formed terms Mα of type α

1 Any variable xα or constant cα is a term.2 If xα is a variable and Mβ a term then (λx .M)α→β is a

term.3 If Mα→β

1 and Mα2 are terms, then (M1M2)β is a term.

3 / 24

Intrinsic Typing

Every well-formed term has an intrinsic type, includingvariables

This kind of intrinsic formulation has become rare, but ithas a number of advantages

Supports conventions, such as

In the remainder of this [talk] we assume thatall terms are well-formed according to the abovedefinition.

In a logical framework

tp : type.

arrow : tp -> tp -> tp.

tm : tp -> type.

lam : (tm A -> tm B) -> tm (arrow A B).

app : tm (arrow A B) -> tm A -> tm B.

4 / 24

Intrinsic Typing

Every well-formed term has an intrinsic type, includingvariables

This kind of intrinsic formulation has become rare, but ithas a number of advantages

Supports conventions, such as

In the remainder of this [talk] we assume thatall terms are well-formed according to the abovedefinition.

In a logical framework

tp : type.

arrow : tp -> tp -> tp.

tm : tp -> type.

lam : (tm A -> tm B) -> tm (arrow A B).

app : tm (arrow A B) -> tm A -> tm B.

4 / 24

Intrinsic Typing

Every well-formed term has an intrinsic type, includingvariables

This kind of intrinsic formulation has become rare, but ithas a number of advantages

Supports conventions, such as

In the remainder of this [talk] we assume thatall terms are well-formed according to the abovedefinition.

In a logical framework

tp : type.

arrow : tp -> tp -> tp.

tm : tp -> type.

lam : (tm A -> tm B) -> tm (arrow A B).

app : tm (arrow A B) -> tm A -> tm B.

4 / 24

Intrinsic Typing

Every well-formed term has an intrinsic type, includingvariables

This kind of intrinsic formulation has become rare, but ithas a number of advantages

Supports conventions, such as

In the remainder of this [talk] we assume thatall terms are well-formed according to the abovedefinition.

In a logical framework

tp : type.

arrow : tp -> tp -> tp.

tm : tp -> type.

lam : (tm A -> tm B) -> tm (arrow A B).

app : tm (arrow A B) -> tm A -> tm B.

4 / 24

Untyped λ-Calculus

(Church 1932) (Church and Rosser 1936)

Terms

1 Any variable x or constant c is a term.2 If x is a variable and M a term then (λx .M) is a term.3 If M1 and M2 are terms, then (M1M2) is a term.

5 / 24

Extrinsic Typing

(Curry 1934) [for combinators, not λ-terms]

Types as properties of terms

Typing judgments defined by rules

x :α ∈ Γ

Γ ` x : α

c :α ∈ Σ

Γ ` c : α

Γ, x :α ` M : β (x 6∈ dom(Γ))

Γ ` λx .M : α→ β

Γ ` M : α→ β Γ ` N : α

Γ ` M N : β

6 / 24

Extrinsic Typing

(Curry 1934) [for combinators, not λ-terms]

Types as properties of terms

Typing judgments defined by rules

x :α ∈ Γ

Γ ` x : α

c :α ∈ Σ

Γ ` c : α

Γ, x :α ` M : β (x 6∈ dom(Γ))

Γ ` λx .M : α→ β

Γ ` M : α→ β Γ ` N : α

Γ ` M N : β

6 / 24

Types as Properties

A term can have multiple types

· ` λx . x : ι→ ι· ` λx . x : (ι→ ι)→ (ι→ ι)

Express explicitly in the form of a single type

Parametric polymorphism (universal types)

· ` λx . x : ∀t. t → t

Ad hoc polymorphism (intersection types)

· ` λx . x : (ι→ ι) ∧ ((ι→ ι)→ (ι→ ι))

7 / 24

Types as Properties

A term can have multiple types

· ` λx . x : ι→ ι· ` λx . x : (ι→ ι)→ (ι→ ι)

Express explicitly in the form of a single type

Parametric polymorphism (universal types)

· ` λx . x : ∀t. t → t

Ad hoc polymorphism (intersection types)

· ` λx . x : (ι→ ι) ∧ ((ι→ ι)→ (ι→ ι))

7 / 24

Types as Properties

A term can have multiple types

· ` λx . x : ι→ ι· ` λx . x : (ι→ ι)→ (ι→ ι)

Express explicitly in the form of a single type

Parametric polymorphism (universal types)

· ` λx . x : ∀t. t → t

Ad hoc polymorphism (intersection types)

· ` λx . x : (ι→ ι) ∧ ((ι→ ι)→ (ι→ ι))

7 / 24

Types as Properties

A term can have multiple types

· ` λx . x : ι→ ι· ` λx . x : (ι→ ι)→ (ι→ ι)

Express explicitly in the form of a single type

Parametric polymorphism (universal types)

· ` λx . x : ∀t. t → t

Ad hoc polymorphism (intersection types)

· ` λx . x : (ι→ ι) ∧ ((ι→ ι)→ (ι→ ι))

7 / 24

Extrinsic Rules

Parametric polymorphism

Γ ` M : β (t 6∈ ftv(Γ))

Γ ` M : ∀t. β∀I

Γ ` M : ∀t. α

Γ ` M : [β/t]α∀E

Intersection polymorphism

Γ ` M : A Γ ` M : B

Γ ` M : A ∧ B∧I

Γ ` M : A ∧ B

Γ ` M : A∧E1

Γ ` M : A ∧ B

Γ ` M : B∧E2

8 / 24

Extrinsic Rules

Parametric polymorphism

Γ ` M : β (t 6∈ ftv(Γ))

Γ ` M : ∀t. β∀I

Γ ` M : ∀t. α

Γ ` M : [β/t]α∀E

Intersection polymorphism

Γ ` M : A Γ ` M : B

Γ ` M : A ∧ B∧I

Γ ` M : A ∧ B

Γ ` M : A∧E1

Γ ` M : A ∧ B

Γ ` M : B∧E2

8 / 24

Types as Properties: The Good News

Can type terms more generally

· ` λx . x x : (∀α. α→ α)→ (∀β. β → β)· ` λx . x x : ((ι→ ι) ∧ ι)→ ι

Can type type terms more accurately

· ` λx . s(s(s x)) : (even→ odd) ∧ (odd→ even)

9 / 24

Types as Properties: The Good News

Can type terms more generally

· ` λx . x x : (∀α. α→ α)→ (∀β. β → β)· ` λx . x x : ((ι→ ι) ∧ ι)→ ι

Can type type terms more accurately

· ` λx . s(s(s x)) : (even→ odd) ∧ (odd→ even)

9 / 24

Types as Properties: The Bad News

The typing judgment is undecidable

Challenge: generalize to a complete language

PracticalUsefulPhilosophically justifiedEasy to reason about

10 / 24

Types as Properties: The Bad News

The typing judgment is undecidable

Challenge: generalize to a complete language

PracticalUsefulPhilosophically justifiedEasy to reason about

10 / 24

Layering Type Systems

Intrinsic simple types for basic consistency

Avoiding Russell’s paradox

Extrinsic sorts on well-formed terms for precision

Circumvent problems with general extrinsic typesAchieve pragmatic goals

Instances

Intersection types to datasort refinement (this talk)Dependent types to index refinements

Parametric polymorphism is a different story

11 / 24

Layering Type Systems

Intrinsic simple types for basic consistency

Avoiding Russell’s paradox

Extrinsic sorts on well-formed terms for precision

Circumvent problems with general extrinsic typesAchieve pragmatic goals

Instances

Intersection types to datasort refinement (this talk)Dependent types to index refinements

Parametric polymorphism is a different story

11 / 24

Layering Type Systems

Intrinsic simple types for basic consistency

Avoiding Russell’s paradox

Extrinsic sorts on well-formed terms for precision

Circumvent problems with general extrinsic typesAchieve pragmatic goals

Instances

Intersection types to datasort refinement (this talk)Dependent types to index refinements

Parametric polymorphism is a different story

11 / 24

Layering Type Systems

Intrinsic simple types for basic consistency

Avoiding Russell’s paradox

Extrinsic sorts on well-formed terms for precision

Circumvent problems with general extrinsic typesAchieve pragmatic goals

Instances

Intersection types to datasort refinement (this talk)Dependent types to index refinements

Parametric polymorphism is a different story

11 / 24

Representing Data

Multiple techniques in Church’s Type Theory

Church numeralsConstants and axioms

We have only basic type ι (o is for truth values)

Example: natural numbers

Constants zι and sι→ι (constructors)Constant natι→o (predicate)Axioms

nat(z)∀x ι. nat(x) ⊃ nat(s(x))

Lists, trees, etc. all have type ι

Sort out using sorts

12 / 24

Representing Data

Multiple techniques in Church’s Type Theory

Church numeralsConstants and axioms

We have only basic type ι (o is for truth values)

Example: natural numbers

Constants zι and sι→ι (constructors)Constant natι→o (predicate)Axioms

nat(z)∀x ι. nat(x) ⊃ nat(s(x))

Lists, trees, etc. all have type ι

Sort out using sorts

12 / 24

Representing Data

Multiple techniques in Church’s Type Theory

Church numeralsConstants and axioms

We have only basic type ι (o is for truth values)

Example: natural numbers

Constants zι and sι→ι (constructors)Constant natι→o (predicate)Axioms

nat(z)∀x ι. nat(x) ⊃ nat(s(x))

Lists, trees, etc. all have type ι

Sort out using sorts

12 / 24

Representing Data

Multiple techniques in Church’s Type Theory

Church numeralsConstants and axioms

We have only basic type ι (o is for truth values)

Example: natural numbers

Constants zι and sι→ι (constructors)Constant natι→o (predicate)Axioms

nat(z)∀x ι. nat(x) ⊃ nat(s(x))

Lists, trees, etc. all have type ι

Sort out using sorts

12 / 24

Simple Sorts

Examplenatι sortzι : natsι→ι : nat→ nat

Define sorts Sα refining type α under signature Σ

1 A base sort Qι declared in Σ is a simple sort.2 If Sα and T β are simple sorts, then (S → T )α→β is a

simple sort.

13 / 24

Simple Sorts

Examplenatι sortzι : natsι→ι : nat→ nat

Define sorts Sα refining type α under signature Σ

1 A base sort Qι declared in Σ is a simple sort.2 If Sα and T β are simple sorts, then (S → T )α→β is a

simple sort.

13 / 24

Sorting Judgment

Context Γ consisting of declarations xα : Sα

Sorting judgment Γ ` Mα : Sα

Defined only for terms of intrinsic type α and sort refiningthe same type α!

Rulesx :S ∈ Γ

Γ ` x : S

c :S ∈ Σ

Γ ` c : S

Γ, x :S ` M : T (x 6∈ dom(Γ))

Γ ` λx .M : S → T

Γ ` M : S → T Γ ` N : S

Γ ` M N : T

14 / 24

Sorting Judgment

Context Γ consisting of declarations xα : Sα

Sorting judgment Γ ` Mα : Sα

Defined only for terms of intrinsic type α and sort refiningthe same type α!

Rulesx :S ∈ Γ

Γ ` x : S

c :S ∈ Σ

Γ ` c : S

Γ, x :S ` M : T (x 6∈ dom(Γ))

Γ ` λx .M : S → T

Γ ` M : S → T Γ ` N : S

Γ ` M N : T

14 / 24

Sorting Judgment

Context Γ consisting of declarations xα : Sα

Sorting judgment Γ ` Mα : Sα

Defined only for terms of intrinsic type α and sort refiningthe same type α!

Rulesx :S ∈ Γ

Γ ` x : S

c :S ∈ Σ

Γ ` c : S

Γ, x :S ` M : T (x 6∈ dom(Γ))

Γ ` λx .M : S → T

Γ ` M : S → T Γ ` N : S

Γ ` M N : T

14 / 24

Subsorts

Subsort declarations Q ι1 ≤ Q ι

2

New rules

Q ≤ Q

Q1 ≤ Q2 Q2 ≤ Q3

Q1 ≤ Q3

Γ ` M : Q Q ≤ Q ′

Γ ` M : Q ′

Defined on base types only

Can derive principles for higher types

15 / 24

Subsorting Example

Refining natural numbers

zero ≤ natpos ≤ natz : zeros : nat→ pos

Examples

· ` λx . x : nat→ nat· ` λx . x : zero→ nat· ` λx . λy . x y : (nat→ zero)→ (zero→ nat)· ` λx . s x : nat→ pos

16 / 24

Subsorting Example

Refining natural numbers

zero ≤ natpos ≤ natz : zeros : nat→ pos

Examples

· ` λx . x : nat→ nat· ` λx . x : zero→ nat· ` λx . λy . x y : (nat→ zero)→ (zero→ nat)· ` λx . s x : nat→ pos

16 / 24

Combining Properties

Want to express and exploit multiple properties of terms

Example: even and odd numbers

even ≤ natodd ≤ natz : evens : even→ odds : odd→ even

Have no way to express in one sort:

· ` λx ι. s(s(s x)) : even→ odd· ` λx ι. s(s(s x)) : odd→ even

17 / 24

Combining Properties

Want to express and exploit multiple properties of terms

Example: even and odd numbers

even ≤ natodd ≤ natz : evens : even→ odds : odd→ even

Have no way to express in one sort:

· ` λx ι. s(s(s x)) : even→ odd· ` λx ι. s(s(s x)) : odd→ even

17 / 24

Combining Properties

Want to express and exploit multiple properties of terms

Example: even and odd numbers

even ≤ natodd ≤ natz : evens : even→ odds : odd→ even

Have no way to express in one sort:

· ` λx ι. s(s(s x)) : even→ odd· ` λx ι. s(s(s x)) : odd→ even

17 / 24

Intersection Sorts

Define sorts Sα refining types α (in intrinsic style)

1 A base sort Qι declared in Σ is a sort.2 If Sα and T β are sorts, then (S → T )α→β is a sort.3 If Sα and Tα are sorts then (S ∧ T )α is a sort.4 >α is a sort for each type α.

18 / 24

Extended Sorting Judgments

Recall Γ ` Mα : Sα

New typing rules

Γ ` M : S1 Γ ` M : S2

Γ ` M : S1 ∧ S2∧I

Γ ` M : S1 ∧ S2

Γ ` M : S1

∧E1

Γ ` M : S1 ∧ S2

Γ ` M : S2

∧E2

Γ ` M : > >I

Philosophically justified in the sense ofDummett/Martin-Lof

19 / 24

Extended Sorting Judgments

Recall Γ ` Mα : Sα

New typing rules

Γ ` M : S1 Γ ` M : S2

Γ ` M : S1 ∧ S2∧I

Γ ` M : S1 ∧ S2

Γ ` M : S1

∧E1

Γ ` M : S1 ∧ S2

Γ ` M : S2

∧E2

Γ ` M : > >I

Philosophically justified in the sense ofDummett/Martin-Lof

19 / 24

Extended Sorting Judgments

Recall Γ ` Mα : Sα

New typing rules

Γ ` M : S1 Γ ` M : S2

Γ ` M : S1 ∧ S2∧I

Γ ` M : S1 ∧ S2

Γ ` M : S1

∧E1

Γ ` M : S1 ∧ S2

Γ ` M : S2

∧E2

Γ ` M : > >I

Philosophically justified in the sense ofDummett/Martin-Lof

19 / 24

Example Revisited

Can now conjoin properties

· ` λx ι. s(s(s x)) : (even→ odd)∧ (odd→ even)∧ (nat→ pos)∧ . . .

Every (well-formed) term has a principal sort

20 / 24

Example Revisited

Can now conjoin properties

· ` λx ι. s(s(s x)) : (even→ odd)∧ (odd→ even)∧ (nat→ pos)∧ . . .

Every (well-formed) term has a principal sort

20 / 24

Some Results

Sort checking is decidable

“Proof:” there are effectively only finitely manyrefinements of a given type

Sorting is closed under β-reduction

“Proof:” standard substitution property

Sorting is closed under η-expansion

“Proof:” induction over sorts

21 / 24

Some Results

Sort checking is decidable

“Proof:” there are effectively only finitely manyrefinements of a given type

Sorting is closed under β-reduction

“Proof:” standard substitution property

Sorting is closed under η-expansion

“Proof:” induction over sorts

21 / 24

Some Results

Sort checking is decidable

“Proof:” there are effectively only finitely manyrefinements of a given type

Sorting is closed under β-reduction

“Proof:” standard substitution property

Sorting is closed under η-expansion

“Proof:” induction over sorts

21 / 24

More Results

Define ηα(M) as η-long form of Mα

Define subsorting Sα ≤ Tα at higher types

S ≤ T :⇔ x :S ` ηα(x) : T

Extend subsumption rule

Sorting is closed under β-expansion

“Proof:” intersect all sorts the abstracted term is used at

Sorting is closed under η-reduction

“Proof:” by subsorting at higher types

ConclusionExtrinsic (Curry) sorting with intersectionsrefining intrinsic (Church) typing is closed underλ-conversion!

22 / 24

More Results

Define ηα(M) as η-long form of Mα

Define subsorting Sα ≤ Tα at higher types

S ≤ T :⇔ x :S ` ηα(x) : T

Extend subsumption rule

Sorting is closed under β-expansion

“Proof:” intersect all sorts the abstracted term is used at

Sorting is closed under η-reduction

“Proof:” by subsorting at higher types

ConclusionExtrinsic (Curry) sorting with intersectionsrefining intrinsic (Church) typing is closed underλ-conversion!

22 / 24

More Results

Define ηα(M) as η-long form of Mα

Define subsorting Sα ≤ Tα at higher types

S ≤ T :⇔ x :S ` ηα(x) : T

Extend subsumption rule

Sorting is closed under β-expansion

“Proof:” intersect all sorts the abstracted term is used at

Sorting is closed under η-reduction

“Proof:” by subsorting at higher types

ConclusionExtrinsic (Curry) sorting with intersectionsrefining intrinsic (Church) typing is closed underλ-conversion!

22 / 24

More Results

Define ηα(M) as η-long form of Mα

Define subsorting Sα ≤ Tα at higher types

S ≤ T :⇔ x :S ` ηα(x) : T

Extend subsumption rule

Sorting is closed under β-expansion

“Proof:” intersect all sorts the abstracted term is used at

Sorting is closed under η-reduction

“Proof:” by subsorting at higher types

ConclusionExtrinsic (Curry) sorting with intersectionsrefining intrinsic (Church) typing is closed underλ-conversion!

22 / 24

More Results

Define ηα(M) as η-long form of Mα

Define subsorting Sα ≤ Tα at higher types

S ≤ T :⇔ x :S ` ηα(x) : T

Extend subsumption rule

Sorting is closed under β-expansion

“Proof:” intersect all sorts the abstracted term is used at

Sorting is closed under η-reduction

“Proof:” by subsorting at higher types

ConclusionExtrinsic (Curry) sorting with intersectionsrefining intrinsic (Church) typing is closed underλ-conversion!

22 / 24

More Results

Define ηα(M) as η-long form of Mα

Define subsorting Sα ≤ Tα at higher types

S ≤ T :⇔ x :S ` ηα(x) : T

Extend subsumption rule

Sorting is closed under β-expansion

“Proof:” intersect all sorts the abstracted term is used at

Sorting is closed under η-reduction

“Proof:” by subsorting at higher types

ConclusionExtrinsic (Curry) sorting with intersectionsrefining intrinsic (Church) typing is closed underλ-conversion!

22 / 24

Related Developments (my students only)

Expressive power extends tree automata to higher types

Canonical (= β-normal, η-long) terms can by typedbidirectionally

In: Festschrift in Honor of Peter B. Andrews on his 70thBirthday, C. Benzmuller, C. Brown, J. Siekmann, and R.Statman, editorsCrucial for logical frameworks (Lovas 2010)

Refinement type inference for ML (Freeman 1994)

Practical refinements type for SML (Davies 2005)

Dependent refinements over decidable domains (Xi 1998)

Unifying sort and dependent refinements (Dunfield 2007)

23 / 24

Conclusion

Church’s original intrinsic formulation of the simply-typedλ-calculus has fallen into disfavor, perhaps unjustly

It suggests an elegant layering with Curry’s extrinsictyping judgment (translated to λ-calculus)

Can be usefully combined with Coppo et al.’s intersectiontypes for high expressiveness, precision, and surprisinglystrong metatheoretic results

24 / 24

Conclusion

Church’s original intrinsic formulation of the simply-typedλ-calculus has fallen into disfavor, perhaps unjustly

It suggests an elegant layering with Curry’s extrinsictyping judgment (translated to λ-calculus)

Can be usefully combined with Coppo et al.’s intersectiontypes for high expressiveness, precision, and surprisinglystrong metatheoretic results

24 / 24

Conclusion

Church’s original intrinsic formulation of the simply-typedλ-calculus has fallen into disfavor, perhaps unjustly

It suggests an elegant layering with Curry’s extrinsictyping judgment (translated to λ-calculus)

Can be usefully combined with Coppo et al.’s intersectiontypes for high expressiveness, precision, and surprisinglystrong metatheoretic results

24 / 24